On the zeros of a class of generalised Dirichlet series-XIV

Abstract

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A. Let 0<θ<1/2 and let {a_n} be a sequence of complex numbers satisfying the inequality |∑^N_m=1a_m − N| ≤ (1/2 − θ)⁻¹ for N = 1,2,3,...,also for n = 1,2,3,...,let α _n be real and |α_n| ≤ C(θ) where C(θ) > 0 is a certain (small)constant depending only on θ. Then the number of zeros of the function ∑^N_n=1a_n (n + α_n)^−s = ζ (s) + ∑^∞_n=1 (a_n(n + α_n)^−s − n^−s) in the rectangle (½−δ ≤ δ ≤ ½+δ,T ≤ t ≤ 2T) (where 0 < δ < 1/2) is ≥ C(θ,δ)T logT where C(θ,δ) is a positive constant independent of T provided T ≥ T₀(θ,δ) a large positive constant. Theorem B. In the above theorem we can relax the condition on a _n to |∑^N_m=1a_m − N| ≤ (½ −θ)⁻¹ N^φ and |a_N| ≤ (½−θ)⁻¹. Then the lower bound for the number of zeros in (σ ≥ ½−δ,T ≤ t ≤ 2T) is > C(θ,δ) Tlog T(log logT)⁻¹. The upper bound for the number of zeros in σ ≥ ½ +δ,T ≤ t ≤ 2T) is O(T) provided ∑_n≤xa_n = x + O_s(x²) for every ε > 0.